# Are all (real) inner products symmetric bilinear forms?

Given that an inner product on a real vector space $$V$$ is a function $$b : V \times V \rightarrow \mathbb{R}$$ satisfying:

$$b$$ is bilinear (that is, $$b$$ is linear in the first variable when the second is kept constant and vice versa); and

$$b$$ is positive definite, that is, $$b(v, v) \geq 0$$ for all $$v \in V,$$ and $$b(v, v)=0$$ if and only if $$v=0$$,

is it true, given this definition, that all such inner products on a real vector space consist of symmetric bilinear forms? I would be grateful for a proof of this, should it be true.

Many thanks

• Can you prove it for one dimensional vector spaces? – Somos May 9 at 10:22